In previous work, confidence regions for general set-identified parameters
were considered by [1]} and [2]}.
Our Theorem 2 is based on their results, especially on Theorem 2.2
of [2]}. [4]}
and [5]} study the asymptotic distribution
of the optimal value in convex-concave minimax problems but do not
consider inference for the solutions of minimax problems. [6]}
consider the GAN problem assuming a single unique solution exists
and give a result on the asymptotic normality of the generator network
parameters; however, they remark (pp. 1560–1561) that the assumed
uniqueness is “hardly satisfied in the high-dimensional context
of (deep) neural networks” and call for generalizations.
Theorem 2 builds on these previous results and provides the first inference
procedure for GAN parameters in the presence of multiple solutions.
Our construction of the confidence sets is specifically tailored for the case of multiple solutions; in the case of a single solution,
confidence sets can be formed based on the results of [6]}
(for details, see Appendix B).
